In [1]}, we showed what additional structure a category needed to be equipped with
in order that it arise from an Ehresmann semigroup.
In Section 2 of this paper, we shall prove Theorem REF , which describes a different (though, obviously, equivalent) way,
in which this can be accomplished.
It arose from reading [2]}.
In Section 3 of this paper, we shall characterize the Ehresmann monoids of the form \(\mathsf {P}(C)\) ,
where \(C\)  is a finite category, in Theorem REF .
This result uses ideas first described in [3]}.
In Section 4, we prove two theorems, as Theorem REF  and Theorem REF , about special kinds of Ehresmann semigroups called restriction semigroups.
